# Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$

Let $$\Phi$$ be a Youngs's function, i.e. $$\Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$$ for some $$\varphi$$ satifying

1. $$\varphi:[0,\infty)\to[0,\infty]$$ is increasing
2. $$\varphi$$ is lower semi continuous
3. $$\varphi(0) = 0$$
4. $$\varphi$$ is neither identically zero nor identically infinite

and define the Luxemburg norm of $$f:\Omega\to\mathbb{R}$$ as $$\lVert f \rVert_{L^{\Phi}} := \inf \left\{\gamma\,\middle|\,\gamma>0,\,\int_{\Omega} \Phi\left(\frac {\lvert f(x)\rvert}{\gamma} \right)\,\mathrm{d}x\leq 1\right\}.$$

Question: What can we say about $$\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$$? In particular, I'd like to know, if $$\Phi\left(\lVert f \rVert_{L^{\Phi}}\right) \leq C \int_{\Omega}\Phi(\lvert f(x)\rvert) \,\mathrm d x$$ holds for some $$C$$ independent of $$f$$.

Any idea or hint for a reference is welcome!

Notes:

• The above inequality trivially holds for $$\Phi(t) = t^p$$, where $$p>1$$
• Maybe it's appropriate to consider this question in the more general framework of Musielak-Orlicz spaces. However, e.g. in Lebesgue and Sobolev Spaces with Variable Exponents I was unable to find an appropriate result.
• Since there has been no response yet, I also asked the question on MathOverflow.
• Do you mean to write $\inf \{\int_\Omega\dots | \gamma>0\}$ instead of $\inf \{\gamma>0 | \int_\Omega\dots \}$? The notation is not clear to me Jun 13, 2019 at 9:02
• No, it's supposed to be the $\inf$ over the $\gamma$. Maybe $\inf\{\gamma\,|\,\gamma>0,\,\int_{\Omega}...\}$ is better? Ah, and I forgot the $\leq 1$ … Sorry! Jun 13, 2019 at 9:26
• A counterexample has been given by @harfe on MathOverflow. Jun 17, 2019 at 6:25